Towards Stable Second-Kind Boundary Integral Equations for Transient Wave Problems

TL;DR

This paper establishes stable discretization methods for second-kind boundary integral equations related to the wave equation in 1D, providing theoretical stability analysis and numerical validation for the first time.

Contribution

It introduces the first stable boundary element method discretizations for second-kind operators in wave problems, supported by comprehensive numerical analysis.

Findings

Boundary integral formulation is $L^2$-elliptic and inf-sup stable.

First BEM discretizations with guaranteed stability for wave equations.

Numerical experiments confirm theoretical stability and accuracy.

Abstract

In this paper, we discuss the stable discretisation of the double layer boundary integral operator for the wave equation in $1d$. For this, we show that the boundary integral formulation is $L^2$-elliptic and also inf-sup stable in standard energy spaces. This turns out to be a particular case of a recent result on the inf-sup stability of boundary integral operators for the wave equation and contributes to its further understanding. Moreover, we present the first BEM discretisations of second-kind operators for the wave equation for which stability is guaranteed and a complete numerical analysis is offered. We validate our theoretical findings with numerical experiments.